We now come to study the two-fold degenerate phonon band crossing point (see Fig. REF (a)) along the Y-\(\Gamma \)  path in R3. \(k_z=0\)  plane is an invariant subspace of \(\widetilde{M_z}\) , and the Bloch states on it can be characterized by its eigenvalues \(g_z=\pm e^{{ik_x}/{2}}\) . According to the Kramer-like degeneracy, we know that the \(k_i=\pi \)  \((i=x,y,z)\)  plane has double degeneracy. At the Y point, which lies on plane \(k_y=\pi \) , one has \([\widetilde{M_z},(\mathcal {T} \widetilde{C_{2y}})]=0\)  and \(g_z=\pm 1\) , so that \(\lbrace |g_z=1\rangle ,|g_z=1\rangle \rbrace \)  are degenerate at this point. In contrast, at the P point (see Fig. S1), a generic point along X-S, lying on the \(k_x=\pi \)  plane, one has \([\widetilde{M_z},(\mathcal {T} \widetilde{C_{2x}})]=0\) , but \(g_z=\pm i\) , such that \(\lbrace |g_z=+i\rangle ,|g_z=-i\rangle \rbrace \)  are degenerate. Consequently, from P to Y, there must be a doublet switching (see in Fig. S1), leading to a neck crossing point of the hourglass-like dispersion. Due to the presence of \(\widetilde{M_z}\) , such a crossing point is not isolated but inducing a line within the \(k_z=0\)  plane, as shown in Fig. REF (b). The shape of the nodal line in the \(k_z=0\)  plane is shown in Fig. REF (b) using black lines. Moreover, we selected a series of symmetry points, a\(_1\) -a\(_4\)  (b\(_1\) -b\(_4\) ), between the \(\rm {\Gamma (Y)}\)  and X (S) points to show the detailed phonon dispersions along the a\(_m\) -b\(_m\)  (m = 1, 2, 3, 4) paths. The results are shown in Fig. REF (c); along the a\(_m\) -b\(_m\)  paths, a series of phonon band crossing points appear, and they belong to the neck crossing points of the hourglass-like dispersion [1]}, [2]}, [3]}, [4]}, [5]}. Therefore, one can see that the two-fold degenerate nodal line phonons in the \(k_z=0\)  plane should be HWNL phonons. Normally, the hourglass nodal line (HNL) can move in the \(k_z=0\)  plane; however, the HWNL as exhibited in Fig. REF (b) is unmovable, as its two endpoints are pinned at the S point. Moreover, as shown in Fig. REF (c), one can see that the hourglass-type phonon band crossings (see orange circles) are almost flat in frequency.
